Selected Topics in Harmonic Maps (Cbms Regional Conference Series in Mathematics)

Conference Board of the Mathematical Sciences

REGIONAL CONFERENCE SERIES IN MATHEMATICS supported by the

National Science Foundation

Number 50

SELECTED TOPICS IN HARMONIC MAPS

by

JAMES EELLS AND LUC LEMAIRE

Published for the

Conference Board of the Mathematical Sciences

by the

American Mathematical Society

Providence, Rhode Island

CONTENTS Introduction........................................................................................................................................... 1

Part L Differential Geometric Aspects of Harmonic Maps.........................................................

~L05,

35J60,

3

1.

Operators on vector bundles...........................................................................................

3

2.

Harmonic maps .................................................................................................................. 13

3.

Some properties of harmonic maps ............................................................................... 21

4.

Second variation of the energy ....................................................................................... 27

5.

Spheres and the behavior of the energy ....................................................................... 32

6.

The stress-energy tensor ................................................................................................... 38

7.

Harmonic morphisms ........................................................................................................ 41

8.

Holomorphic and harmonic maps between almost Kiihler manifolds ..................... 47

9.

Properties of harmonic maps between Kiihler manifolds .......................................... 53

Part II. Problems Relating to Harmonic Maps.............................................................................. 63

ane Univer·

1­

82·25526

1.

Existence of harmonic maps ........................................................................................... 63

2.

Regularity problems .......................................................................................................... 66

3.

Holomorphic and conformal maps ................................................................................. 68

4.

Construction/classification of hamlonic maps ............................................................. 70

5.

Properties of harmonic maps .......................................................................................... 72

6.

Spaces of maps .................................................................................................................. 74

7.

Noncompact domains ....................................................................................................... 76

8.

Variations on a theme ...................................................................................................... 76

Bibliography for Part 1........................................................................................................................ 79

Supplementary bibliography for Part I1. .......................................................................................... 83

t.

•lishers.

v

Introduction The fITst part of this work is devoted to an account of \arious aspects of the theory of harmonic maps between Riemannian manifolds. In § 1 we develop the formalism of Rie­ mannian connections in vector bundles and the relevant calculus of vector bundle valued dif­ ferential forms. That formalism is applied systematically in the sequel. § § 2-7 give a rather full treatment of various topics, as indicated by their titles in the list above. §§8 and 9 pre­ sent certain aspects of the relationships between harmonic and holomorphic maps. Our primary aim in Part I is to present a coherent introduction to harmonic maps as a branch of geometric variational theory, and to illustrate their appearance as significant ob­ jects in Riemannian geometry. The title "Selected topics" indicates that important aspects of the theory of harmonic maps are not included in our exposition; in particular: the essential theory of existence and regularity, boundary value problems, the case of noncompact manifolds, the relationship with other variational prinCiples, the recent applications to the theory of Kahler manifolds, to three-dimensional topol· ogy, and to the study of curvature. For overall information, guidelines and an extended bibliography, we refer

to

(18] and

(4IJ. In Part II we propose certain unsolved problems, together with comments and refer­ ences. They range over the whole theory of harmonic maps, and are certainly of widely varying difficulty. The present text grew from lectures given at Tulane University in December 1980. In addition to topics recorded here, the lectures included an account of a classification theorem for harmonic maps of a two-sphere into real and complex projective spaces. That theory­ which has developed rapidly in the past few months is only briefly summarized in the notes and comments on §9, and will be the object of a memoir of J. Eells and J. C. Wood. ACKNOWLEDGEMENTS. The first·named author thanks the National Science Foundation for support of the Regional Conference at Tulane University at which he delivered the lec­ tures from which the present text has evolved. The second-named author is Research Asso­ ciate at the National Fund for Scientific Research (Belgium) and was also supported by the Sonderforschungsbereich Theoretische Mathematik (SFB 40) of the University of Bonn.

2

JAMES EELLS AND LUC LEMAIRE

Special thanks are due to Professor R. J. Knill, his organizing committee consisting of A. L. Vitter, M. Kalka, and P.·M. Wong, and their Administrative Assistant Ms. Jackie Boling. Their efficient handling of the administration and cheerful good will made this Regional Con­ ference a pleasure to all participants. The present text has benefited from comments by and conversations with the partici­ pants at the conference, as well as P. Baird, F. E. Burstall, B. FugIede, C. M. Wood and J. C. Wood. We hereby record our thanks to all of them. Special thanks are due to Secretary Mrs. P. Q. Lam for typing this manuscript. CONVENTIONS. M and N denote metrizable smooth (Le., COO) connected manifolds of dimensions m and n, which (for simplicity of exposition) we suppose orientable and without boundary. They are endowed with smooth Riemannian metrics g and h, with respect to which they are complete. Unless otherwise specified (in such a form as "M being a complete manifold...") we also suppose that M and N are compact. All maps (x). (1.4) DEFINITION. A Riemannian metric on a vector bundle V is a section a in 2 C(0 V'll) which induces on each fibre a positive defmite inner product. We shall often write a( a, p) = (a, p), even using the same ( , ) for different bundles ap­ pearing in the same calculation, or (a, p)a when confusions could occur. (1.5) DEFINITION. A linear connection on a vector bundle ~: V - M is a bilinear map v on spaces of sections:

v:

C(TM) x C(V) -

a) f-;o vX a, X E C(TM), a E C(V) and such that for IE C(M), we have a (0 VfX = Iv xa, (ii) vxU' a) = XI' a + Iv x a.

vxa is called the covan'ant derivative 01 a in the direction of X.

(1.6) If a and bare metrics on V and W respectively, we can induce metrics on the

written

v: (X,

C(V),

bundles of (1.3) as follows:

3

4

JAMES EELLS AND LUC LEMAIRE

For V*, we use on each fibre Vx and V; the musical isomorphisms:

~: Vx

#: V;

V; defined by a~' P

----l> ----l>

For a, {3 E

= (a,

p}a. for a, P E Vx and

Vx defined as

V;, we set (a, {3}v' =

,iyt>v'

For a, p E Vx and A, J.l. E Wx ' we set

(a

EEl A, pEEl J.l.>

(a

0 A. p ® J.l.>

this product induces one on For ¢: M

cr

1

----l>

IV and

1]:

= (a, p> + (A, J.l.>

= (a, p) • (A, J.l.) A,P V and 8 P v.

in Vx EEl Wx ' in Vx 0 Wx ;

W ----l> IV a vector bundle with metric b. we can identify

with a, p E W¢-l w (a, P)b' (1. 7) If v v and vware connections on Vand W, we define also:

the dual connection \j* on V* by

a, p E

W)x

(\j ;8)

where 8 E C(V*), a E C(V), the direct sum connection

\j

.a

= X'

(8 . a) - 8 .

\j xa,

on V EEl W by

\j x(a

EEl A)

= \j fa EEl

\j ~A,

the tensor product connection on V 0 W by

These definitions are of course chosen in a way insuring the validity of the expected Leibnitz formulae for the derivative of products.

IV and a vector bundle W ----l> N with connection \j w, we define the pull·back connection on ¢-I W as follows (a suggestion of Professor A. Machado): it is the unique connection \j on ¢ -1 W such that for each x EM, with y = i/J(x) EN, X E TxM and A E C(W), we have For a smooth map ¢: M

----l>

\j x( ¢*A)

where di/J: TxM

----l>

i/J* \j

:1¢. xC A)

TyN is the differential of ¢ and ¢*A

=A

0

i/J E C(¢-I W).

To prove the existence and uniqueness of V, consider on a neighborhood Uy of y in N a frame field (Aa)1 ~a~n' i.e. a system of sections of W providing a base of each fibre. If Ux is a neighborhood of x such that ¢(Ux ) C Uy and if p is a section of ¢-1 Wover Ux ' there exist n functions (fa) on Ux such that p summation symbol on repeated indices.

= fa

. i/J*Aa' where we have omitted the

Imposing that V be a connection satisfying the condition above implies in Ux that

Vx p

= \j xUa • ¢*Aa) = (X . r)i/J*Aa + r (X, r)i/J*A a

+r

. i/J*(v:i'XAa)'

. Vx i/J* Aa:

5

HARMONIC MAPS I}

must therefore be unique, and it is straightforward to check that this formula is in­

dependent of the choice of frame field and defmes the required connection.

Itify

(1.8) DEFINITION. A Riemannian structure on a bundle V is a pair (I}, a), where a is a Riemannian metric, I} a connection and I}a == 0, where I}a is defmed as in (1.7). The condition I}a 0 means that for each X E C(TM), 0, p E C(V), we have X· (0, p) == (I}xo, p) + (0, I}x p )· It is straightforward to check that if (I} v, a) and (I} w, b) are Riemannian structures on V and W respectively, then the metrics and connections built in (1.6) and (1.7) form Riemannian structures on the different bundles under consideration. (1.9) EXAMPLE. On the tangent bundle TM, the. torsion of a connection I} is defined by T(X, y)=-l}xY+ l}yX+ [X, Y]

for X, Y E C(TM). If g is a Riemannian metric on TM, the fundamental theorem of Riemannian geometry asserts that there is one and only one connection (the Levi-Civita connection) such that I}g == 0

Indeed,

-g(X, [Y, ZJ)

on 'i}w, we ~.

Machado):

)EN,

T == O.

is defined and characterized by the formula

2g(1} xY, Z) == X· g(Y, Z) +

expected

l

I}

and

y. g(Z,

X) -

z· g(X,

+ g(Y,

[Z, X])

+ g(Z,

y) [X, Y]),

for all X, Y, Z E C(TM). Note that on other vector bundles, the torsion is not defmed and a metric does not usually determine a unique connection. In all our applications of this formalism to harmonic maps, we shall consider connec­ tions induced from the Levi-Civita connections on tangent bundles by the constructions of

Uy of yin N

(1.7).

n fibre. If

In particular, for a smooth map rp: M --+ N, we shall repeatedly consider on rp -1 TN the pull-back 1}tJ;-1 TN of the Levi-Civita connection on N. We now note the following

over Ux ' 1itled the

formula: if X, Y E C(TM), then

u.>: that Indeed, consider around the points x EM and y == rp(x) EN two systems of coordi­ nates (Xi) and (uOt such ~)

=

Now 'iJT*M i dx j

a/ax

_MrZ!kdxk and

'-Lagrange 1/>.

so that ,,)O< ( 'iJ d 'I' Ij -

,,0< _ Mrk"o< + Nro' = O. (2.7) EXA..'APLE. The identity map I: M, g - - M, g is trivially harmonic. However, we shall see in (4.19) that it is far from being innocuous from the point of view of the be­ havior of the energy. (2.8) EXAMPLE. If N = R, a harmonic map f: M, g -- R is a harmonic function. In that situation, one can also define subharmonic and superharmonic functions as follows: (2.9) DEFINITION. A real valued function f: M, g -- R is subharmonic iff I::..f':;;;' O. It is superharmonic iff t:...f;;. o. Recall the sign convention: I::..f == trace 'iJd[. We shall use the following (2.10) MAXIMUM PRINCIPLE. If U is a domain of M, g and f: M, g -+ R a subhar­ monic function having a maximum at an interior point of U, then f is constant. See e.g. [26, Chapter 3] . (2.11) DEF1NITION. For a map 1/>: M, g -- N, h, the quadratic form 'iJdl/> is called the second fundamental form of the map 1/>.

(2.12) PROPOSITION. For X, Y E C(TM), 'iJdI/>(X, Y)= 'iJ~-ITNdl/>' Y-dq,(\I~Y).

1.20) that

Indeed, \ldq,(X, y) == (\I xdl/»Y = \If 1TN(dl/> . Y) - dq,('iJ1fY).

d (uO is symmetric, i.e. for all X, Y E C(TM), 'iJdq,(X, Y) \ldrt>(Y, X).

:ymbols of I order to

PROOF. Since \1 M and \IN are torsionless, we have \ldcp(X, y) - \ldq,(Y, X)

= \I~-lTN(dl/> . Y) -

\I~-l TN(dcp • X) - drt>(\I~Y - \I¥X)

= dl/> . [X, Y] - dl/> . [X, Y]

=0

by (1.9).

16

JAMES EELLS AND LUC LEMAIRE

This shows in particular that vd¢ is indeed a quadratic form at each point x EM, since for

X, Y E C(TM) and f E C(M), we get vd¢(jX, Y) = (v fxd¢)Y = fvd¢(X, Y) and vd¢(X, fY) = vd¢(fY, X) = fvd¢(X, Y). (2.14) COROLLARY. For any ¢ E C(M, N), the Horm d¢ E

Al(¢-lTN) is closed,

i.e. dd¢ = O. This is a reformulation of Corollary (2.13), since d is the antisymmetrization of (1.16).

v by

A map ¢: M, g ---'> N, h is harmonic iff drp is a harmonic I-form

(2.15) COROLLARY.

with values in rp-l nv. Indeed,

rp is harmonic iff d*drp = O. Since ddrp = 0 and M is compact, this is equiva­

lent to t.d¢ = 0 by (1.24). (2.16) EXAMPLE. Suppose that ¢: M, g

N, h is a Riemannian immersion. Identi­ fying X E C(TM) with drp . X E C(rp-1TN), we see that v~Y = v~Y + vd¢(X, Y). Comparing with [34, Vol. II; VII, 3], we see that the connection on N is decomposed in its tangential component (the connection on M) and its normal component (the second fundamental form in the classical sense). (2.17) DEFINITION. The mean curvature of the immersion is the trac;;Of the second / fundamental form divided by m = dim M. We recall that the immersion is minimal iff its mean curvature vanishes, and get im­

mediately:

---'>

(2.18) PROPOSITION. A Riemannian immersion is harmonic iff it is minimal. Let us now go back to the case of a map (2.19) DEFINlTlON.

rp: M, g ~ N, h.

rp is totally geodesic iff vd¢ = O. This name will be justified

shortly, by means of the following composition law, which will also yield many applications in the next section. (2.20) PROPOSlTlON. If(M, g), (N, h) and (P, k) are three manifolds and ¢ E C(M, N), 1/1 E C(N, P), then

vd(1/I

0

rp) = d1/l

0

vdrp

+ vd1/l(drp,

r(1/I

0

rp) = d1/l

0

r(rp)

+ trace

vd1/l(d¢, drp).

rp and 1/1 are totally geodesic, so is totally geodesic, then 1/1 rp is harmonic. In particular, if

drp),

1/1

0

rp, and if rp is harmonic and

1/1

0

It is important to note, however, that the composition of two harmonic maps is not harmonic in general.

T

J i7

HARMONIC MAPS :: M, since for

PROOF.

) and

'Vd(1/I

0

¢>)(X, Y) == 'Vx(dl/l . d¢>' Y) - d(1/I

= ('V dcp' xdl/l )d¢> . Y + dl/l rN) is closed,

0

¢». 'VxY

. 'V xed¢> . Y) - dl/l . d¢> . 'V xY

'Vdl/l(d¢> . X, d¢> . Y) + dl/l . 'Vd¢(X, Y). Taking traces yields the second formula.

ation of 'V by

(2.21) PROPOSITION. The following three properties are equivalent: (a) ¢> is a totally geodesic map. (b) ¢> preserves connections. (c) ¢> maps geodesics of M linearly to geodesics of N.

'1armonic I-form

PROOF. The equivalence (a) this is equiva­

'Va/ax

¢(X, Y). is decomposed

It (the second

yf the second

(b) follows from the formula in Proposition (2.12).

Suppose then 'Vd¢> = O. Then for any geodesic path ,: (-e, e) -

lersion. Identi­ o



M, we have

de¢> 0 1) dl (d1 d1)_ dx =d¢>'lJa/axdx+lJd¢>dx'dx =0.

Conversely, if the image of each geodesic is a geodesic with proportional parametriza­ tion, we get 'Vd¢>(dl/dx, d,/dx) == 0 for each vector d,/dx, so that 'Vd¢> = O. (2.22) We now examine the relation between a Riemannian immersion ¢>: M, g ___ Rn

, and get im­

and its Gauss map 1: M, g - G(n, m), and prove a theorem of E. Ruh and J. Vilms [52]. Consider the mean curvature of ¢>, i.e., (l/m)trace 'Vd¢> = (l/m}r(¢». It is a section of ¢>-l TRn which is normal to the image of N, so that it can be viewed as a section of the nor­

ninimal.

mal bundle V(N, M). Its covariant derivative in that bundle is defined as the projection in n V(N, M) of its derivative in 1 TR , and we shall denote it by 'Vl«(I/m)trace 'Vd¢». (2.23) DEFINITION. ¢> has constant mean curvature if'V l «(1/m)trace 'Vd¢» O.

r

This condition implies in particular that 1(1/m)trace IJd¢>1 is constant, but the converse

ill be justified any applications

md IjJ E C(M, N),

is not true for n - m ;;;. 2, as we shall see in the example of (2.30) below.

(2.24) DEFINITION. G(n, m) is the Grassmannian manifold of m-spaces through the origin in Rn. The Gauss map 1: M - G(n, m) associated with the immersion ¢>: M ___ R n assigns to the point x EM the m-space tangent to ¢(M) at ¢(x), translated to the origin of Rn. Recall (see, e.g., [34, II], [49]) that if P is an m-space through the origin of Rn and P its associated point in G(n, m), the tangent space of G(n, m) at P can be realized as the space of linear mappings from P to its orthogonal complement. If K denotes the bundle on G(n, m) whose fibre at

P is P,

we have therefore TG(n, m)

= K*

® Kl.

A point P of G(n, m) being given, we can choose an orthonormal frame (e l ,

""

en)

n

larmonic and 1/1 ric maps is not

of R such that (e l , " " em) is a basis of P and (em + 1"'" en) a basis of pl, Then the canonical Riemannian structure on G(n, m) is defmed by requiring that et ® er

(i

= 1,

"" m, r

m + 1, ... , n) be an orthonormal basis of TpG(n, m),

Another interpretation of the tangent space can be given as follows. For the basis chosen above, represent P by the m-vector e l 1\ ... 1\ em' Then et ® er can be identified

18

JAMES EELLS

AND

LUC LEMAIRE

with the m-plane

(2.25) THEOREM [521. Let ¢: M, g - Rn be an isometric immersion. Then the tension field of the Gauss map 'Y can be identified with the covariant derivative in the normal bundle of m times the mean curvature of ¢: 1'('Y) = vl1'(¢). Therefore, ¢ has constant mean curvature iff'Y is harmonic. PROOF. Around a point Xo EM, consider a system of normal coordinates (xi). In these coordinates, 'Y maps a point x onto o¢/ox 1 " ••• " o¢/oxm, and the differential of 'Y is given by

Choosing an orthonormal basis e l ' d¢ . %xi1xo' we get at xo:

... ,

en of Rn such that for i = 1, ... , m we have e/

=

Indeed, the coordinates xl are normal and Rn is flat. Now the second fundamental form of an immersion has only normal components, denoted by hij = [Vd1f>(%x i , %x i )]" so that

f. = r=m+l =

f

htje 1

"

•• , "

ej _ 1

"

er "

ej+l "

.••

"em

j=1

L: L: h~jE;' r

j

On the other hand, we have

Vd" (x. ,:,) ~ (VxdOl(,:,) ~ '/(X)V,/"C:,)

01

Vd,,(,:,)l.o ~ hffet 0 e,.

Using the above identification, we see that d'Y = Vd¢, where d'Y E C(T*M ® 'Y- 1 TG(n, m» and Vd¢ E C(@2T*M® VeRn, M». Through this identification the two bundles under consideration are isometric and one can check that they have the same connection, so that we have where Vi is the connection in @2 T*M ® V(Rn , M). Viewing Vd¢ as a section of @2 T* M ® 1 TRn , we have then V1 Vd¢ = (VVd¢i, the projection on V(N, M) of VVd¢. We shall take the trace of the above equality, using the notation trace V_V_ to indicate that the trace is taken on the two marked vectors. We get

r

HARMONIC MAPS

19

1"(r) == trace vdr == (trace v_ v_d¢/f

== (trace v_ vdlj)(_))l == (trace(vv_dlj)(-) + R( ,-)dlj)(_»)l, Then the n the normal nstant mean : (Xi). In :rential of 'Y

the curvature being that of T*M® lj)-lTRn , i.e., minus that of 1M. Therefore,

r('y) == v l trace vdlj)

+ (dlj) . R M (-,

)_)1

= vlr(lj» + O.

(2.26) COROLLARY. A Riemannian immersion lj): M, g fundamental form iff its Gauss map is totally geodesic.

R n has parallel second

Indeed, we have seen that vd'Y vlvdlj). (2.27) We can now give another interpretation of the constancy of the mean curvature, due to Matsushima [44]. Consider vdlj) as a one-form with values in T*M ® V(N, M). We get

(2.28) PROPOSITION. Let lj): M, g - Rn be a Riemannian immersion, and consider vdlj) E Al(T*M ® V(N, M). Then (0 d(vdlj» = 0 is satisfied and is Codazzi's equation. (ii) d*(vdr/» = 0 iff M has constant mean curvature.

=

ental form of 1:1)] r, so that

Indeed, d(vdlj» vtv 1dlj), where the bracket denotes antisymmetrization, and this quantity is the left-hand side of Codazzi's equation [34, Vol. II, (VII, 4)]. By (1.20), d*(vdlj» = -trace v~ \/_dr/> == - v l trace vdlj) by the calculations leading to Theorem (2.25). (2.29) EXAMPLE. Let Sn denote the Euclidean sphere and r/>: M, g - Sn a Rieman­ nian immersion of constant mean curvature. Let I{;: Sn - R n + 1 be the canonical embed­ ding. Then I{; 0 r/> has constant mean curvature. PROOF. By (2.20), we have to calculate \/1r(1{; 0 r/» == \/l(dl{; . r(lj» + trace \/dl{;(dr/>, dlj»).

Now

'Y-1TG(n, m» mdles under

by (2.16). The first term is zero by hypothesis and the second vanishes because dQ>(X) and r(r/» are orthogonal, using the special form of vdl{;. If (e j ) denotes an orthonormal basis at a point of M and Il the outer normal vector Il, so that field to Sn, we see for each i that vdl{;(dlj) . e j , dr/> . ei )

=-

!ction, so that

tion of Q!i T* M 'vvdr/>. We shall ate that the trace

Therefore, vlr(1{; 0 r/» == O. (2.30) REMARK. Let lj): M, g -

N, h be a minimal immersion and I{;: N, h _ R n +1 an immersion of constant mean curvature. It is not true in general that I{; 0 lj) has constant mean curvature.

20

JAMES EELLS AND LUC LEMAIRE

Indeed, consider the simple case of a helix on a cylinder C of R 3 . The map cp: R ----+ C defining the helix is a geodesic (in fact a linear map) and the cylinder has constant mean cur­ vature. In coordinates x on Rand (u l , u 2 , u 3 ) in R 3 , l/J 2 3 (U 1 ,U ,U )

(cosx,sinx,a'x),

0

cp is given by aER,

and we have

r(l/J

0

cp)

Va!axr(l/J

0

cp):= (sin x, -cos x, 0).

:=

(-cos x, -sin x, 0),

This vector is tangent to the cylinder, but not to the helix, for a =F O. Therefore,

V~/oxr(l/J 0 cp) =F O. Note on the other hand that Ir( l/J 0 cp)1 is constant. We can also observe that the Gauss map of l/J 0 cp maps x on _ (

sin x ~,

)1

+ a2

i.e. on a circle of S 3 which is not a geodesic for a =F O. Therefore, the Gauss map is not harmonic.

Notes and comments. (2.31) As an immediate application of Corollary (2.15) and the third remark of (1.28), we have [60]:

Let M, g be the Euclidean or hyperbolic m-space (m ;;;. 3). Let cp: M, g ----+ N, h be a harmonic map of finite energy. Then cp is constant. In contrast, it was shown by Sacks-Uhlenbeck [53] that for m := 2, any harmonic map cp: R2 ----+ N with E(1)) < 00 has a unique extension to a harmonic map 75: S2 ----+ N (2.32) Although the composition of two harmonic maps is not harmonic in general, various supplementary hypotheses lead to composition laws. For example, suppose that

M p

N

1 1 rr

P~Q

is a commutative diagram, where p and

1T

are Riemannian submersions, with l/J *TH M C THN.

where T;;M is the orthogonal complement of ker dp(x). Assume that one of the following conditions is satisfied:

(a) l/J *(TM) C TH N, (b)

has totally geodesic fibres, (c) for all z EP, p l(Z) -;. 1T-l(¢(Z)) is a Riemannian fibration with minimal fibres. 1T

Then r( 1T

1T * r(1/1), so that 1T

l/J is harmonic iff r( l/J) is vertical. A special case of this statement is used in [23] (see (9.33) for a statement of some re­ 0

l/J)

:=

sults from that paper).

0

r

I 21

HARMONIC MAPS

Ie map ¢;: R ~ C

:mstant mean cur­

(2.33) Lawson [37] has constructed minimal immersions in S3 of oriented Riemann surfaces of all genera. By (2.29) these induce immersions of constant mean curvature in R 4 , whose Gauss maps are harmonic by (2.25). Since the Grassmannian GO(4, 2) of oriented two-planes in R4 is isometric to S2 x S2, this provides examples of harmonic maps from these surfaces to S2 (see [19] for details). (2.34) Harmonic maps can be characterized in probabilistic terms. Putting aside tech­ nical details, the situation is the following: The submartingales on R are those stochastic processes expressible in the form Z

+ A,

where Z: £o(R) x R (> 0) ~ R is Brownian motion on R starting at 0 E R, possibly with fefore,

a random time change, and A is a continuous increasing process on R. A stochastic process

X:

n

x R (> 0) -

M on a Riemannian manifold is a martingale on M, g if for any convex function f: U - R defined on an open subset U of M, the composition f 0 X1u is a sub­ martingale on R. Brownian motion ZM: £a(M) x R (;;;;. 0) ~ M, g starting at a EM is an

example. Then a map ¢;: M, g-N, his

lSS

map is not

I remark of (1.28),

. g-N, h bea my harmonic map N.

')2 _

.onic in general, suppose that

(i) harmonic iff ¢; 0 ZM is a martingale on N, h, (ii) a harmonic morphism in the sense of § 7 below iff ¢; 0 ZM = ZN 0 a, where a is a suitable random time change. Apparently, the Euclidean case of (ij) goes back to P. Levy. We learned of 0), in a different form, from [46]. (2.35) If M, g and N, h are pseudo-Riemannian and ¢; E C(M, N), one can define as above the energy and tension of ¢;, and extremals of E will still satisfy the system r(¢;) = O• However, E can take negative values and r( ¢;) = 0 is not in general elliptic, so that the exis­ tence theory will be very different, and is essentially unknown at present. We mention how­ ever a recent paper of Gu [10], who-motivated by the Lorentzian metrics of physics and the relations between harmonic maps, a-models and Yang-Mills fields-studied a Cauchy problem for harmonic maps from R2 with signature (1,1) to a Riemannian manifold. Note also that if M, g is pseudo-Riemannian and N. \l is an affine manifold, the system 1(1/» = 0 is still defined for maps ¢;: M - N. However, the energy of I/> is not defined in that case.

3. Some properties of hannonic maps. h !J;*THM C THN, of the following

(3.1) We first present two properties of harmonic maps which will appear here as im­ mediate consequences of similar statements proven in § 1 for harmonic forms. (3.2) UNIQUE CONTINUATION THEOREM [54]. Let ¢;: M. g ~ N, h be a harmonic

map which is constant on an open subset ofM. Then

I/>

is constant on M.

th minimal fibres.

71.

ement of some re-

Indeed, dl/> is a harmonic one-form by (2.15), and is zero on the open set, so that Theorem (1.26) implies that it is zero everywhere. The following Weitzenbock formula is crucial in the paper of Eells-Sampson [20]:

f

! 22

JAMES EELLS Ai\lD LUC LEMAiRE

(3.3)

PROPOSITION.

- Trace

If 0). The concave side of the surface is the side pointed at by the components Vdi(X, X). If for instance Sn is a sphere in R n + I , the concave side is the interior of the sphere.

t.

e its image is

(3.10)

Let S be a hypersurface 01 (N, h) and Yo E S a point at which Vdl is definite. Then there is a neighborhood V ofYo in N and a strictly convex function 1 I: V ---+ R such that (0) = S n V and f < 0 on the concave side of S. LEMMA.

r-

In a neighborhood V of Yo in N, take a system of local coordinates un) such that S n V is given by un = 0, the coordinate lines of un are geodesics

PROOF.

(u

l

, ...•

normal to S, and un measures the distance to S, with a negative sign on the concave side and a positive sign on the other. With the induced coordinates (Xl, ... , x n - l ) on S n V, we see that I: (Xl, ... , x n - 1 ) f-> (Xl, ... , x n - l , 0) and un: (u l , ... , un) ---+ un, so that

un

0

i = O. Thus the composition formula (2.20) yields

rc and

'ormula of that

Id4>1 2 is

o

dun

0

vdt

+ Vdun(di,

Since un is negative on the concave side, we get for all X E TyOS, X*' 0,

Vdun(di . X, di' X) = -dun the rank of rJ> t rJ> is totally

di).

0

Vdt(X, X)

> O.

However, un is not strictly convex, for Vdun(o n'o n) = 0, but it suffices to set (for inu u stance) I (un + 1)2 - 1 in a possibly smaller neighborhood to obtain a function with the

*' X

tlyone.

required properties. Indeed, for 0

e now intro­

for au n we get Vdl(o u n' 0 u n)

,he manifold

Let N, h be a complete manifold with RiemN ~ O. II S is a closed totally geodesic submanlfold of N, then the function Is: N ---+ R given by

(3.11)

E TyOS, we still have vdf(dl . X, dt . X)

> 0 and

= 2.

PROPOSITION.

lposition for-

where p is the distance lunction, is smooth and convex on a geodesic tubular neighborhood Vol Sin N (i.e., a tubular neighborhood whose fibres are geodesic balls). II S is a point, fs is strictly convex.

24

JAMES EELLS A.1\fD LUC LEMAIRE

We follow Bishop-O'Neill [7]. For each y E V denote by c y the shortest

PROOF.

geodesic segment from y to S. c y is then perpendicular to S and is unique. We parametrize it in such a way that it is represented by the map cy : [0, 1] -+ N, mapping 0 to y and 1 to S. Then

Now y -+ -c~(1) is the inverse of the exponential map restricted to the normal bundle of Sin N, so that y -+ !c~(1)!2 is a smooth function. Let 0'* X E TyN and denote by t -+ bet) = expitX) the geodesic segment starting at y with initial velocity X. Define C: I x 1-+ N by C(x, t) = cb(t)(x), Le. the geodesic starting from bet) to end perpendicularly on S. We shall then calculate the second variation of the energy of that geodesic with respect to the variation X. The formula is of course classical in this case, but as we shall prove it for all harmonic maps in the next section, we shall start here from formula (4.2) below, which yields

Integrating the first term of the right-hand side by parts gives

1( il a/ atat oC OC) I - f.0 , ila/ax ox dx

t=O

X

oC OC)I =l! + I\ il a/ atar, ox _

x -0 t=O

.

The first expression vanishes since C(x, 0) is a geodesic. At x = 0, C(O, t) is also a geodesic so that the second term vanishes for x

= O.

Finally, for x:::: I, the path t ---+ C(I, t) lies in

the totally geodesic submanifold S, so that il afa toqot/ x = oqox/x = l ' Finally, there remains only

~ ildfs(y)(X, X) =

= The curvature Suppose we must have x I, C(1, t)

=

oqot!x =0

2 d2 E (C(x, -

dt

t))

is tangent to S and normal to

I t=O

OCI2 f.l/ N(OC OC)oC OC)I f. ll ila/axatt=odxo\R oX'Of ox'ot t=odX. 0

hypothesis insures that ildfs(YXX, X) ;;;. O. now that S is a pOint and let X,* O. Suppose that ildfs(Y)(x, X) = O. Then ila/axoqot = 0 along t = 0, i.e. oqot is parallel along C(x, 0). But for is constant since S is a point, so that oC/ot :::: O. By parallel transport we get

= X = 0, contradicting the assumption.

(3. I 2)

1

COROLLARY.

Therefore, fs is strictly convex.

If N, h is simply connected, the above proposition is valid on

the whole of N. We can now draw a number of conclusions on the behavior of harmonic maps.

HARMONIC MAPS wrtest ,z,ametrize y and 1

25

(3.13) PROPOSITION [28]. Let M, g be a compact manifold and N, h a complete wm/old. If ¢: 11;£ ~ N is a harmonic map whose image is contained in a domain V of N carrying a strictly convex function!. then ¢ is constant. PROOF.

By the composition law (2.20), we have - fj,(f ¢) = trace 'iJdf(d¢, d¢) ;;;. O.

bundle of

By the maximum principle (2.10), we must have f



¢ constant, so that

If ¢ were not constant, we could find a vector X such that 'iJdf(d¢ . X, d¢ . X) mt starting geodesic ad variation ,f course ;ection, we

¢) = O. > 0, con­

fj,(f

tradicting this equality. (3.14) EXAMPLES. (i) Any point of a manifold N has a neighborhood carrying a

strictly convex function (Lemma (3.8».

(ii) Any complete simply connected manifold with Riem N vex function (Corollary (3.12». (iii) A noncompact complete manifold with Riem N

,;;;;

0 carries a strictly con­

> 0 carries a strictly convex

func­

tion (Greene-Wu); and so does an open half-sphere. (3.15) PROPOSITION. Let M, g be compact and N, h complete with Riem N dX\

t=O

.

,;;;;

0 in a

geodesic tubular neighborhood V of a closed totally geodesic submanifold SeN. Then any harmonic map ¢: M, g ~ N, h with ¢(M) C V and ¢(M) n S ¢ has image entirely in S.

'*

PROOF.

The function fs: V

~

R of Proposition (3.11) is convex, so that fs

0

subharmonic and therefore constant. But there is a point Xo EM for which fs(¢(xo)) so that fs ¢ =: O.

¢ is

= 0,

o

The following characterizations are due to T. Ishihara:

uso a geodesic

(3.16) THEOREM [33]. A map ¢: M, g ~ N, his (a) totally geodesic iff it carries germs of convex functions to germs of convex func­

C(1, t) lies in

d normal to

tions;

(b) harmonic iff it carries germs of convex junctions to germs of subharmonic func­ tions.

PROOF OF (a). For f a function on a domain V of N, we shall start from the compo­

sition law 'iJd(f ¢) = df 'iJd¢ + 'iJdf(d¢, d¢). If 'iJd¢ = 0 and f is convex, we get

'iJd(f ¢)(xo)(v, v) = 'iJdf(¢(xo))(d¢ . v, d¢ . v) ;;;. 0

r,

X) = O. Then ). But for transport we get

for all v E TxoM, so that f ¢ is c;)nvex on ¢-l(V). To prove the converse, suppose that ¢ preserves convex functions but that there exist Xo EM and v E TxoM such that w = 0 in T¢(Xo)N. Using Lemma (3.8), we can find in a neighborhood of 'iJd¢(xo)(v, v) Yo = ¢(xo) a convex function f with df(yo)w < -ld¢(x o )vI 2 and 'iJdf(yo) = I. Then

'*

convex. ion is valid on which contradicts the assumption that f lic maps.



¢ is convex.

I

I

I 26

JAMES EELLS AND LUC LEMAIRE

PROOF OF (b). This follows the same lines, starting with the composition law

- tl(f 0 ¢) == df 0 T(I/»

+ trace

vdf(d¢, dl/».

°

If ¢ is harmonic and f convex on V C N, we get -tl(f 0 1/» ;;;. on rl(V) so that f 0 I/> is subharmonic. Conversely, if at a point Xo EM, T(I/>Xx o ) = w 0, choose a convex function f around Yo = ¢l(xo) with df(yo)w < -trace!d¢l(x o )1 2 and Vdf(yo) == 1. This gives -tl(fo I/»(x o) < 0, again contradicting the hypothesis. The following maximum principle is due to Sampson [54J; we follow the proof of

*'

[70J. (3.17) THEOREM. Let 1/>: M, g

~

N, h be a nonconstant harmonic map and SeN a hypersurface with definite second fundamental form at a point Yo = ¢l(xo) E S. Then no neighborhood of Xo EM is mapped entirely in the concave side of S in N. PROOF.

f defmed in a neighbor­ on the concave side of Sand f- 1 (0) n V = S n V.

Use Lemma (3.10) to fmd a strictly convex function

°

hood V of Yo in N, such that f < Now if I/> maps a neighborhood U of Xo to the concave side of S in V, then for all x E U we

°

have f(if>(x)) ~ = f(¢l(x o )), so that Xo is a maximum. But the composition formula shows that - tl(f 0 1/» ;;;. on U, so that f 0 I/> := 0. Therefore tl(f 0 1/» 0, and since f is strictly convex, I/> must be constant. Notes and comments. (3.18) As an analogue of Corollary (3.4), we have [56J: If Mis noncompact and RicciM ;;;. 0, Riem N ~ 0 and 1/>: M, g ~ N, h is a harmonic map of finite energy, then I/> is constant. (3.19) A further study of the formula of Proposition (3.3) yields the following result ([61] , also [18 J for a special case): Let 1/>: M, g ~ N, h be a harmonic map between compact Riemannian manifolds such that there exist strictly positive constants A and B with Ricci M ;;;. A . I and Riem N ~ B. Suppose that rank I/> ~ p and e(l/» ~ (pj2(p - l))(A/B). Then either I/> is constant or s, t denotes the differential along TM (and not along T(M x R x R));

is a harmonic As in the proof of Proposition (2.4), we see that

following result

manifolds such Riem N 0;;;;; B. Istant or rjJ is a whose /ift to a weak one im­

I

-

'0 ,'iJa/atd¢s,t ) 'iJ -a - w >• s s,t=O


rems of the RN (d . X d .

1.) == RN (d . X'at'0
i.e. if u is a Jacobi field in the classical sense.

u)¢' == 0;

= 0 iff

(

I

•

29

HARMONIC MAPS

M - N be harmonic and suppose Riem N ~ O. Then and index = 0 by (4.3).

(4.10) EXAMPLE.

Hrp(l),

I)

;;;;.

0 for all

I)

Let

:

In fact, much more is known in this case: Hartman has shown that a homotopy class contains precisely one connected subset of harmonic maps, all of minimum energy [30], which has been further described by Schoen·Yau as a manifold with nonsmooth boundary

[57]; see

also [66]. (4.11) When no restriction is made on the curvature of N, an important observation is

that very simple harmonic maps have positive index and are therefore not minima of E. This is of course a serious obstacle for existence theory. Following R. T. Smith

[65],

we first consider the simplest possible map, namely an

isometric diffeomorphism from M, g to M, g. By using that map to identify the two mani· folds, we can reduce it to the identity map I: M, g

~

M, g.

1M is simply a vector field on M and JI) - trace vM vM I) - Ricci( I). Using musical isomorphisms, we shall identify the vector field I) with the one· form I)~, and use this to define the differential, codifferential and Laplacian of I) as

A section

I)

of

11*1)

= d*I)~,

(4.12) Warning. We now have two Laplacians acting on form

)perator, and

I)

with values in 1M, and

w>

- (trace (t::/ v, w)

v2 v, w> - L,(RM(e.S' w)v, es> + (Ricci v,

w)

where (e s ) is an orthonormal basis at the point under consideration. Using (4.4), we deduce that

ltroduce are

Jlv)

(4.13) ,pace of n of the kernel

acts on the real valued one·form I)~. By Weitzen·

s

fferential oper· -1 TN) splits real numbers,

1:/ which acts on the zero·

bock's formula (1.34), they are related by (~I),

and consisting

Li" which

I):

= ~v -

2 Ricci(v).

To proceed, we shall use a formula of K. Yano [74] and a well-known characterization of conformal vector fields.

(4.14) LEMMA. For v E C(1M) , up to

)/(os)"

vg =

for

JM(~ILlJgI2 -

(d*V)2) Vg

where Lv g is the Lie derivative of the metric g in the direction of v.

as

PROOF. Around a point x O' we consider a system of normal coordinates and set

=

a/ox s ,

We show at

integral vanishes. We set

Xo

that (Jv, v> - ~ IL v gl 2

+ (I1*V)2

is a divergence, so that its

30

JAMES EELLS AND LUC LEMAIRE

(*) is a divergence, and we shall show that it is equal to the above expression. First %11IU!2 = -% trace vv/u/ 2 == -(trace vvu, u) -lvul 2

= (flu, U)

+ (U' Ot)] = (VOtVOsU' Os)(U, Ot) - (vas VatU, 0s)(U' Ot)

+ (VosU'

Os)(Va t U, Ot) -

= -(Ricci(u), u> + (d*U)2

(vatU,

0s)(VosU' Ot)

- (vatU, os>(vosu, Ot>

so that

Since Lug(X, y) = g(vxu, Y)

%iLugl2 (4.15)

+ g(X,

Vyu), we have finally

= %«Vosu, Ot) + (vatU, = O.

n :?' 3. Then index((oj, oJ = 0,

34

JAMES EELLS AND LUC LZ:VIAiRE

+ ['laid'3 j X} = -(Xf'

v)d¢ . OJ. Since

1 Lx(e . Vg ) 1'>1 (d¢,

== -

fM (7,

'l(d¢ . X)Vg

d¢ . X)Vg -

Since this is true for any X, we get div S,;> == -

(7,

+

I'll

1'>1 ('lX, S¢)Vg (X,

div

Sr/J)vg ,

d¢).

(6.7) COROLLARY. If ¢ is harmonic, then Sq, is conservative (i.e., div S1> = 0). If ¢ is a differentiable submersion almost everywhere and if div Sq, 0, then ¢ is harmonic. Note that in the second statement, the su~ectivity of d¢ is an essential hypothesis. For example, let ¢: M, g --->- N, h be any Riemannian immersion (in general not harmonic). Then S¢ == «m - 2)/2)g and div So == 0 since 'lg == O. Let us now consider some special situations. 1, then Scp = - 161¢ '1 2 , and along a geodesic parametrized propor­ (6.8) If dim M tionally to arc length we have of course 19 'I constant. (6.9) DEFINITION. A map 9: il.f, g --->- N, h is (weakly) conformal if ¢*h == pg, where p E C(M) and p ;;;, O.

m

(6.1 0) PROPOSITION. Let ¢: ill. g 2 and ¢ is conformal. PROOF. If Scp

= 0,

--->-

N, h be a nonconstant map. Then Sq, == 0 iff

then ¢*h == e . g and

o

trace Sri> == e . trace g

=e

. m - 2e

= (m -

trace ¢*h

2)e

so that m = 2. Conversely, if ¢*h == pg, then e = mp/2 and Sq, == «m - 2)/2)pg. This leads to the following observation (which we first learned from J. H. Sampson): (6.11) PROPOSITION. If m ¢ is homothetic.

> 2 and ¢:

M, g

--->-

N, h is harmonic and conformal, then

Indeed, with these hypotheses:

o == so that J1 is constant.

div S¢ ==

m-"J

m

"J

2 - div(pg) = ~(dJ1,

g)

HARMONIC MAPS (6.12) PROPOSITION. Let fjJ: M, g

that

--'Jo

41

lV, h be a totally geodesic map.

Then v(fjJ*h)

== 0, e is constant and vS", == O. PROOF. At a point x o ' let X, Y, Z be three tangent vectors, and extend Y and Z in a neighborhood in such a way that at Xo we have vxY == 0 vxZ. Then, at xo:

)tain

(vxfjJ*h)(Y, Z)::: vx(fjJ*h(Y, Z))

== since vdfjJ

= 0, so

Nowe

I>

= 0).

ItfjJ

:rmonic.

rpothesis. t harmonic).

== vx(dfjJ' Y, dfjJ' Z)

v

< xdfjJ . Y, dfjJ .

Z)

+ (dfjJ

. Y,

vxdfjJ . Z)

0,

that v(fjJ*h) == O.

*(g, fjJ*h) so that

Therefore, vS", = O. We shall also use the stress-energy tensor in subsequent sections. REMARK (ADDED IN PROOF). Theorem (6.3) was also obtained by A. I. Pluznikov in Harmonic mappings of Riemann surfaces and foliated manifolds Math. Sb(N.S.) 113 (1980),

339-347. (Russian) zed propor· 'h

pg, where

en S", = 0 iff

7. Hannonic morphisms. (7.1) DEFINITION. A map fjJ: M, g -- lV, h is a harmonic morphism iff for any har­ monic function f defined on an open set V of lV, the composition fa fjJ is harmonic on fjJ-I(V).

It is clear that the notion of harmonic morphism is purely local; and that the composi· tion of two harmonic morphisms is a harmonic morphism. Our first aim is to present a characterization of harmonic morphisms due to B. Fug. lede [24] and T. Ishihara [33] in terms of the following property. (7.2) DEFINITION. A map fjJ: M, g --'Jo lV, h is horizontally conformal iff for any x such that d¢(x) =1= 0, the restriction of d¢(x) to the orthogonal complement of Ker dfjJ(x) is conformal and surjective. We shall use the following notations: Ker dfjJ(x) = T:M (the vertical space) and (Ker d¢(x»l = T!jM (the horizontal space).

3. Sampson):

(7.3) LEMMA. Let fjJ: V - - W be a nonconstant linear map between Euclidean vector spaces, and fjJ*: W --'Jo V its adjoint map (where V* and W* are identified with Vand W,so

conformal, then

that fjJ* is characterized by (fjJ*(w), v)

== == (2e/n)(X, y>.

Indeed (d¢ . X, d¢ .

= r,2 . (X, Y) and e == nr,2/2.

Y)

(7.5) THEOREM [24,33]. A map ¢: M, g ---;. N, h is a harmonic morphism iff it is a harmonic and horizontally conformal map. If it is nonconstant, it is a submersion on an open dense subset of M, so that m ;;:?; n. If at a point x, rank d¢(x) < n, then d¢(x) == O. The proof given in [33] is based on an extension of a lemma of Bers [6], which we shall state here without proof.

Let Yo EN and consider a system of normal coordinates (u") around Yo' For any system of constants (c", c,,(3) with c"[3 == c[3" and k~==l c,," = 0, there is a harmonic junction f defined in a neighborhood of Yo and such that 3f/3u"lyO == c" and a2J,.f/3u"3u[31Yo = c,,[3' (7.6)

LEMMA.

(7.5). Suppose that ¢: M, g ---;. N, h is a harmonic morphism. For any f: V ---;. R (V eN) with - 6.f == 0, we have by (2.20): PROOF OF THEOREM

(7.7)

-6.(10

1/»

== dfo

7",

+ trace vdf(d1/>, d¢) == O.

For any X o EM, consider a normal chart (u") around ¢(xo ). In a neighborhood of ¢(xo), Lemma (7.6) implies for each 'Y E (1, ... , n) the existence of a harmonic function f such that at ¢(x o), af/au" == Ii",), and a2 f/3u"3u[3 == O. At x O' (7.7) implies therefore that 7~ == 0, so that ¢ is harmonic. We apply the lemma again, this time with CO! == 0 and any c,,{3 = c[3O! such that kC",,,, == and get from (7.7) that

°

3¢" aqll ..

trace vdf(d1/>, d1/» = c",[3-. - ,gil == 0,

,

ax! ax'

or

L

a*[3

i

Ca{31/>t¢Jg i +

L C",,(¢Nt a

By choosing different values for ca{3' we get ¢tY.1/>tg ij so that ¢f¢fli == A2(x)oa{3.

=:

¢l¢J )gii == 0. ¢f¢Jgii and ¢r¢fgii == 0 for a =I=~,

If A(X) = 0, then rank d¢ == 0, If A(X) =1= 0, then (drj!)* is conformal and Lemma (7.3) implies that drj! is horizontally conformal and of rank n. Suppose that ¢ is nonconstant. The set M' eM on which rank d¢ == n is of course open. It is also dense, since if on an open set we had drj! =: 0, then drj! would be zero every­ where by the unique continuation theorem (3.2).

HAR:V10NIC MAPS

43

Conversely, suppose now that I/> is a harmonic and horizontally conformal map, and let

(, YE T;:M

f: V --l- R, V C N, be a harmonic fUnction. For any Xo E 1> -1 V, 'Ne consider systems of normal coordinates (Xi) around Xo and UOl around I/>(x o)' and get

1 'Jrphism iff it is a nersion on an

o

== O.

len dq,(x)

so that

(6] , which we

f

0

¢ is harmonic.

(7.8) REMARK. It appears from this proof and Lemma (7.4) that ¢ is a harmonic (2e(I/»/n)(t::.f) 0 1/>.

morphism iff for each harmonic function f on V C N one has t::.(f 0 1/» lares (U oo ) around = 0, there is a lyO

=

Coo

(7.9)

COROLLARY. If 1/>: ly!, g

--l-

N, h is a Riemannian submersion (i.e., at each

point dl/> is surjective and dl/>ITHM isometric) then the following conditions are equivalent:

and

(a) I/> is a harmonic map, (b) ¢ is a harmonic morphism, (c) the fibres are minimal.

:mic morphism.

The equivalence between (b) and (c) will follow from Proposition (7.18) below, since erp is constant.

We note the following variation in the definition of harmonic morphisms: :ighborhood of nonic function

(7.10) PROPOSITION [24, 33]. A map ¢: M, g --l- N, h is a harmonic morphism iff its composition with sUbharmonic functions is sUbharmonic.

f

ies therefore that PROOF. Assume first that ¢ is a harmonic morphism and let f: V --l- R, V C N, be a subharmonic function. Then

such that

-t::.(fo ¢) = trace Vdf(dl/>, dl/»

-i'?(t::.f)

0

1/>;;;;'

a

so that f 0 n.

We suppose from now on that m

In a neighborhood of any point Xo EM, consider an orthonormal frame field (Xa) 1

48

JAMES EELLS AND LUC LEMA!R!':

T'N and T"N, Thus we define a¢: T'M -- T'N, a¢: TUM -- T'N,

a¢;: T'M -- T"N,

a.p: One checks that a;P

c

0

T"M -- T"N,

= a¢ (the complex conjugate) and a;P = a¢. d ¢IT'M = a¢

J

and

+ a¢-

and

c

By construction,

--­

d ¢IT"M = a¢

+ a¢.

A map ¢ is called holomorphic iff J 0 d¢ = d¢ " J iff a¢ = 0 and antiholomorphic iff = -d¢ 0 J iff a¢ = O. We shall call ± holomorphic a map which is holomorphic or

d¢

antiho]omorphic. (8.4) Using the almost Hermitian structures of M and N, we define the partial energy densities of ¢ as the following squares of complex norms:

where fields.

¢F (resp. ¢~) is the

matrix representation of a¢ (resp. a¢) in the chosen local frame

J

We have e(¢) = e '(¢)

+ e"(¢).

Note that la¢i x is not the Hilbert·Schmidt norm of a¢(x) seen as a real linear map from T;M to T~(x)N, and this justifies the absence of the factor

*

in the definition of e '(¢).

With M compact we set

E"(¢) = JM r e"(¢)vg and obtain E(¢) = E'(¢) + E"(¢). Obviously, ¢ is hoI om orphic iff EI/(, and ker drp can be identified with

d¢/

(kerd1»

.L

em -I.

=I

fold

a or 2. = 0,

dw and

. d¢ . X

Then (ker d1>(x))l ~

e and

conj

commutes with I and is homothetic. By the characterization of Theorem (7.5), (see

1> is a harmonic morphism.

that

(8.18) COROLLARY. Suppose that rpo is harmonic and a minimum of E in its class,

sym

and that it is homotopic to a ± holomorphic map 1>1' Then 1>0 is ± holomorphic. Indeed, if 1>1 is, say, holomorphic, the minimum of E" in the class is 0, and t: M -

N is a smooth deformation of a ± holomorphic map

¢o through harmonic maps the complex bilinear extension of h to reN, we observe in a local chart that

(¢*h)2,0 = dz 2 , ( denotes the complex bilinear extension of h to TeN,

ence a branched

This formula was used by Suzuki [67,68J and Siu-Yau [64], and we shall follow the proof of [64]. Another form of (9.18) has been obtained by T. Ishihara [32].

58

JAMES EELLS Ai'1D LUC L':::MA!::>"E

PROOF.

We have lr

e (¢)

A

= , z z

(9.19) Now we shall need the following two formulae:

wi

sil PROOF OF THE LEMMA. Using the properties of Kiihler manifolds, we shall consider

around a point Yo EN a system of holomorphic normal coordinates, so that at Yo' 0IJ. h _

=

a == 0_v hex,6_ and a'Yo 6 hex,6_ =

O. Using the formulae recalled in (9.1) for the

connectio~,6

fa

tu

and the curvature, we get

('1_¢_)"1 8

Z

m

= AS0_(¢2) + h'YP(o h _)¢::¢~ z ex {Jp S z

so that at the point Wo we have 51

= asa 0 -(¢'2) + h'P.E..(a h _)¢::¢~ s z as" ,6 p 8 2

(V '1_¢_)"1 s s z

Z

p j:

( t

which proves (i). Formula (ii) is established similarly. We now substitute (i) in (9.19) and obtain

Using the divergence theorem and (9.20) (ii), we note that

-J., N is a harmonic map and there is a point Xo E M at which rank d¢ ;;;;, 4 and such that RN (¢(x o)) is strongly negative, then ¢ is ± holomorphic.

[i.e., any ele­

fore,

).

omorphic map

Siu also provided some important refinements and applications of that theorem. For instance THEOREM [63 J. Let M and N be as above with dimcM;;;;' 2 and RN strongly negative. Then any harmonic oriented homotopy equivalence is a biholomorphic diffeomorphism. ApPLICATION. If M and N are compact Kahler manifolds of complex dimension at least two having the same homotopy type, and if RN is strongly negative, or if N is a com­ pact quotient of a classical bounded symmetric domain, then M and N are ± biholomorph­ ically equivalent.

JAMES EELLS AND LUC LEMAIRE

(9.32) Generalizing results of [39, 64J we note the [22J. Let M be a compact Riemann surface and N a simply connected Kahler manifold with 1f 2 (N) generated by a holomorphic map cp l --» N. Then any map 1>: M --» N of minimum energy in its homotopy class is ± holomorphic. PROPOSITION

Some homotopy hypothesis is necessary, for there are K3 surfaces N which are sim­ ply connected Kiihler manifolds for which every holomorphic map 1> : M --» N is constant, and by a theorem of Sacks·Uh1enbeck [53 J maps. As an application, we cite the

1f 2 (N)

is harmonically generated by minimizing

COROLLARY [22]. If W:M --» T2 +n is a conformal immersion of a Riemann surface into a flat torus of dimension at least 3, then its Gauss map 1: M --» Qn == G°(2 + n, 2) has minimum energy iff M = S 2 or Wis a minimal immersion.

(9.33) In very broad terms, here is a statement of the main results of Eells-Wood [23]. Let M be a Riemann surface (open or closed) and 1> : M --» cp n a map into complex projec­ tive n-space. If H --» cp n denotes the Hopf bundle, we can defme a universal lift n?

».

r surfaces N

maps. : maps ~ following : curvature. 1in [86]. 5 to be much of nonexis­ that we have

Certainly every such ± holomorphic map is constant (see Part I (8.21 The case m = n + 1 is especially interesting, for we know the homotopy classification, and it is sim­ ple: [Cpn ... 1 , Cpn] = 0 for n odd,

Z2

for n even.

(1.16) Smith [112] has shown that for m .;;;; 7 every homotopy class of maps Sm -+ Sm contains a harmonic representative. We have no idea whether that dimension restriction is essential. However, in the delightful company of Giusti and Miranda it was found that the limitation of Smith's method was just that which appears as limitation in the solution of the Bernstein problem.

66

JAMES EELLS AND LUC LEMAIRE

flJthough several interesting harmonic maps Sm ~ Sm are known for m

> 7 (see

[18,

is

§8]), no general existence theorem is known in these dimensions. (1.17) Problem. Discuss the existence of harmonic maps Sm

~ sm for m > 7. (1.18) Existence theory is especially important and interesting for domain manifolds

ane to

M with boundary. See [18, § 12; 99 and 115] for an account of the main results obtained to date. In that case, one can consider homotopy classes relative to Dirichlet, Neumann or mixed boundary problems. Apart from the results mentioned above, it is not clear what kind of conditions on, say, Dirichlet data would imply existence or nonexistence of a solu­ tion.

C1 me diff

2. Regularity problems.

(2.1) The fundamental regularity theorem for harmonic maps is that a continuous weakly harmonic map of class L~ is smooth [18,97]. As already noted in (1.3), if dim M == 2 certain existence results can be obtained from a theorem of Morrey to the effect that an L -map from M ~ N, h is smooth, provided it minimizes E over all sufficiently

i

can

an

small disks.

(2.2) Problem. Let M be a closed Riemann surface and if>: M ~ N, h an Li-harmonic map. Is if> continuous? A partial result has been obtained by Gruter [95], who has shown that an Li-harmonic map if> which is weakly conformal (in the sense that (¢*h)2,0 == 0) is continuous. Grilter's proof utilizes a property of approximate differentiability of q-maps from surfaces (due to H. Federer). Problem (2.2) is a special case of the following conjecture of S. Hildebrandt [97, §2], dim which we formulate as (2.3) Problem. Let M be a surface and rp: M, g ~ N, h an q-extremal of a func­ tional

F(if» == f'l1f(x, rf>(x), \lqi(x))vg

is

where f has quadratic growth in its third variable. If F conformally invariant with respect to M, is rp continuous? (2.4) For any manifolds M and N, Schoen-Yau (107) have indicated how an ac­ tion of an Li-map on classes of loops defines a conjugacy class of homomorphisms 'lT (M) ~ 'IT 1 (N), which coincides with the usual class if the map is continuous. Together l with Morrey's regUlarity theorem, this leads to an alternative proof of existence of harmonic maps, provided dim M = 2 and 'lT2(N)

appt

= O.

If dim M;;;' 3 let a be a given conjugacy class of homomorphisms

'IT 1 (M) ~ 'IT 1 (N);

Geo

let F denote the set of Lf-maps inducing a. It follows from [107) that E takes an absolute minimum at some map 0 E F. That led Schoen to pose the

Sch(

strai (2.5) Problem [105]. Is that absolute minimum 0 continuous? If not, is if>o contin­ rp:k uous off a subset of codimension 2? tinuc ADDED IN PROOF. This problem has been solved by Schoen-Uhlenbeck, see (2.10) below. (2.6) Problem. Let

~m

1>: it!, g

C

h(!.>

~ N, h be a continuous Li-map between two manifolds

of the same dimension, and S¢ its stress-energy tensor. Suppose (1) that the Jacobian of if>

The redu

T

! 67

HARMONIC MAPS

, 7 (see [18,

n> 7. manifolds

is defined and positive almost el'erywhere; (2) div So == 0 in a distributional sense. Is ¢ harmonic? It is Li-harmonic? As Sealey has emphasized [59], there is special interest in the case dim M = 2 dim N and (Scl»2,O == _(¢*h)2,o holomorphic, for a positive answer would give a positive solution

ts obtained

to Shibata's problem [109; see §5 below]. The answer is yes if ¢ E C2 [70]; and if ¢ is a

eumann or

C1-diffeomorphism between compact surfaces [59]. We also call attention to the announce­

~ar

ment [108].

what

ADDED IN PROOF.

: of a solu-

Problem (2.6) has been solved affirmatively by Sealey for dim M =

= 2, see (5.2) below. (2.7) Instead of regularity for the second order system defining harmonic maps, we

dim N Iltinuous

can pose the following, which may well be easier:

(2.8) Problem. Let M and N be Kahler (or even Hermitian) manifolds and ¢: M

), if tc the effect

~

N

= O. Is ¢ continuous (and therefore holomorphic)? (2.9) In most discussions of regularity problems, the space L i(M, N) is defined (by

an Li-map such that E"(¢;)

'ficiently

means of an embedding of N in Rq) as

.i-harmonic {E Li(M, Rq): (M) CNa.e.}.

Li-harmonic s. Gruter's

Problem. Is the space C""(M, N) dense in Li(M, N)? Equivalently, is CaCM, N)

n

lces (due to

Li(M, N) dense in q(,tf, N)?

A modification of an example of [98] suggests that the answer could be negative for ldt [97, §2], dim M ~ 3, but says nothing about dim M = 2. ADDED IN PROOF. Problem (2.9) was solved by Schoen-Uh1enbeck (Boundary regu­ of a func­ larity and miscellaneous results on harmonic maps). Using the above-mentioned example, they show that C""'(M, N) is not always dense in Li(M. N) when dim M = 3. On the other hand, they prove that it is dense if dim M = 2. ADDED IN PROOF. with respect (2.10) Important regularity results have been recently obtained: M. Giaquinta and E. Giusti, On the regularity of the minima of variational integrals, how an ac­ Acta Math. 148 (1982),31-46. phisms The singular set of the minima of certain quadratic functionals, Analysis (to . Together appear). ; of harmonic ~ 1f 1 (N);

es an absolute

R. Schoen and K. Uhlenbeck, A regularity theory for harmonic maps, J. Differential Geom. 17 (1982), 307-335. (2.11) Giaquinta-Giusti consider the case of maps from a manifold to Rn, whereas Schoen-Uhlenbeck study maps between two manifolds, realizing the range as a set of con­

straints in a Euclidean space. This put aside, the main result in common is that an L; -map , is ¢o contin­ ¢: M ~ N which minimizes E locally is HOlder-continuous on M\SIj> = {x EM: ¢ is con­ see (2.10)

tinuous in a neighborhood of x}, where the singular set S

satisfies 17'*S2,O

Wood [115,

+ 17"*SI,1

0,

(see Part I (9.7));lor dimcM;;;' 2 do these equations imply sigmficant global restrictions on ¢, as they do in case dimcM = 1 (Part I, (9.11)-(9.14))? . Much pro­

ori estimates

(3.6) The following was suggested by J. C. Wood, and has served as motivation for much in [23]:

(3.7) Problem. If M is a closed Riemann surface of genus M = p and ¢: M --+ Cpn a harmonic map of degree ¢ ;;;. p, is ¢ weakly conformal? As we have already noted, the answer is yes if n = 1, with any metric on Cpl. It is

rv between

also yes if p = 1 [116]. On the other hand, for 0 ~ deg( 0 we know that H contains a harmonic map 9. If ¢ is locally bijective then its Jacobian l¢ 0 on M, from which we can conclude that 9 is a diffeomorphism. (See (5.6) below.) Thus Problem (5.2) is solved if we can show that some harmonic map 9 E H is locally bijective. An important attack on that problem has been made by Shibata [109], with clarifica­ tions and modifications by Sealey [59]. However, we believe that the problem remains open. It has been reduced to the regularity question discussed in (2.6). ADDED IN PROOF. Problem (5.2) has now been solved affirmatively -by Sealey (The regularity of quasiconformal 5-harmonic mappings) who completed Shibata's program, using a theorem of Seretov [108] to solve Problem (2.6) in dimension two. See also H. Sealey, Harmonic diffeomorphisms of surfaces, Harmonic Maps Tulane (1980), Lecture Notes in Math., vol. 949, Springer-Verlag, Berlin and New York, 1982, pp. 140-145; and On the existence of harmonic diffeomorphisms of surfaces (preprint).

i·

'*

73

HARMONIC MAPS

-by 1. lost and R. Schoen (On the existence of harmonic diffeomorphisms between

ps

surfaces, Invent. Math. 66 (1982), 353-359), using a previous result of Jost (Univalency of -;. S4 with

harmonic maps between surfaces, 1. Reine Angew. Math. 324 (1981), 141-153).

1; and

see [88].

= 0 or I)?

(5.3) The following conjecture has been around for several years. Sampson has con­ sidered it, with the hope of finding a proof of Mostow's rigidity theorem via harmonic maps; it has been formally posed by Lawson-Yau.

(5.4) Problem. Let M, g and N, h be compact manifolds with strictly negative curva­ ture, and ¢: M ....... N a harmonic homotopy equivalence. Is ¢ a diffeomorphism?

f a smooth

In general, fibration

The answer is yes if dim M 2 dim N, and in the case of flat manifolds. However, as we have noted in [89], Calabi [84] has given examples of metrics g on the torus T m (m ~ 3) such that a harmonic map of Tm, g to a flat torus T m cannot be a diffeomorphism. (5.5) Problem. Are those maps homeomorphisms? Note that with respect to the local problem, H. Lewy [101] has shown that any harmonic homeomorphism between open sets of R2 is a diffeomorphism. By way of contrast, J. C. Wood [70] has observed that the map ¢: R3 ....... R3 given by (x, y, z) ....... (x 3

-

3xz 2

+ yz,

y - 3xz, z)

is a harmonic polynomial homeomorphism, with lacobian determinant

3x 2 •

m

. The exis­

(5.6) It is well known that if U and V are domains in C and ¢: U""'" V is a holo­ morphic homeomorphism, then its Jacobian determinant =1= 0; in particular, ¢ -1: V ....... U is hoiomorphic. (5.7) Problem. If ¢: m ....... Cm is a complex polynomial map with Jacobian deter­ minant =1= 0, is ¢ biholomorphic?

c

ertain har­ luestion, a n the

Li­

, endowed ism. Does

Bieberbach has given examples when m = 2 to show that such an assertion is not true for hoi om orphic maps, in general. Apparently it is known that in (5.7) we have dim(C m ¢(C m » .;:;; m - 2. See [102, 110]. The next problem is due to Lawson: (5.8) Problem. Let ¢: (Dm+ 1, Sm) ....... (Dm+ 1, Sm) be a harmonic map such that

¢Ism is a homeomorphism. If m .;:;; 5, is ¢ a diffeomorphism?

know that on M, from :oblem (5.2) nth c1arifica­ remains

completed dimension 1aps Tulane York, 1982, es (preprint).

The basic question of rank of harmonic maps has been studied by Sampson [54] : (5.9) Problem. Let ¢: M""'" N be a harmonic map and U an open subset of M such 0 or that rank ¢I u .;:;; k. Then does ¢ have rank ¢ .;:;; k on all ,M? The answer is yes if k k = 1 [54]; and of course, if M and N are both real analytic. This question might be examined in the framework of differential forms: If w is a

harmonic I-form with vector bundle values and wx: TxM""'" Vx has rank';:;; k on U, does w have rank';:;; k on M? Looking more closely at the singular set of ¢ (Le., the set of points of M at which rank ¢ is not maximal), we formulate the (5.10) Problem. Let ¢, l/I be harmonic (resp., holomorphic) maps between Riemann­ ian (resp., Kdhler) manifolds. If they have the same singular structure-in a sense to be made precise-do they essentially coincide?

74

JAMES EELLS AND LUC LEMAIRE

(5.11) J. C. Wood described completely the possible singularities of harmonic maps between surfaces [70j. For a map I/> with nonvanishing Jacobian of a closed Riemann sur­ face M of genus p to a flat torus, he showed that I/> has at most 2p 2 branch points,

(2p - 2)(6p - 6) general folds,

6p - 6 meeting points of general folds.

(5.12) Problem. Are these bounds attained? Are there similar bounds for maps onto a surface of higher genus, with a metric of constant negative curvature?

de: thE (p(

Such a restriction on the metric of the range is necessary, for without it a construction of R. T. Smith could produce harmonic maps with arbitrarily many folds.

(5.13) J. C. Wood has shown that for a nonconstant harmonic map 1/>: M, g -+ N, h

tria

between compact real analytic surfaces,

f

q,(JIf)

KNvh ;;. 21TX(I/>(M»,

where KN denotes the Gauss curvature of N [70j. He asks:

(5.14) Problem. Is there an analogous inequality for higher dimensional range? (5.1 5) Problem. What sort of Runge approximation theorem can be established for harmonic maps? For instance, let K be a compact subset of a complete manifold M, and 1/>0 a harmonic map of a neighborhood of K into N. Wben can we approximate 1/>0 uniformly on K by a harmonic map 1/>: M -+ N? We should expect topological restrictions on M - K (e.g., that M - K should have no compact components). See [81 J for the linear case. (5.16) Problem. Let K be a compact subset of M and 1/>: M - K -+ N a harmonic map. Under what conditions can we assert that I/> has an extension to a harmonic map 1/>: M -+ N? For instance, if D is a 2-disk and 1/>: D {O} -+ N a harmonic map of finite energy, then I/> extends to a harmonic map 1/>: D -+ N [53j. On the other hand, the analogous as­ sertion for m-disks (m ;;. 3) is false without further growth restrictions at O. See [18; 10.15, 12.10J. In general, we should expect restrictions on the capacity or Hausdorff measure of K. (5.17) Problem. Calculate the index of harmonic maps 1/>: Sm -+ Sn. If m;;' 3, we have seen in Part I (5.17) that index(l/» ;;. max . rank ¢; + 1. See also

[59J for special maps. (5.18) Problem. Let M be a closed Riemann surface of genus p and ¢;: M -+ CpI a harmonic map. What is the value of index(¢;)? If ¢; is holomorphic, and in particular if degree(¢;);;' p. then its index is zero. For 0< degree I/> < p - 1, on the other hand, nonholomorphic examples exist [40J, and by [23] . their index is ;;. (degree( 0) + 1 - p) for degree I/> ;;. p /2. 6. Spaces of maps.

into clos

han sal ( Rier:

ham exiSI

Ph. J tence. ;;. 3, gies.

also. , necte is of· Lie gl

(6.1) Let M, N be compact. The space H(M, N) of harmonic maps is locally com­ pact and locally finite dimensional.

carrie! gree p

75

HARMONIC MAPS

,nic maps :'11ann sur­

(6.2) Problem. Is HOII, N) an absolute neighborhood retract? If RiemN ,.;;;; 0 and C is a component of C(1);!, N). then H ::::: C() H(fvi, IV) is a compact deformation retract of C. Therefore the answer is yes in that case. Also, for any point a EM the evaluation map eVa: H -+ N is an immersion onto a totally geodesic submanifold of N (possibly with Lipschitz boundary); see [66,57]. Furthermore,

r maps onto

1TlH) =

°

1T 1 (H)

centralizer of ¢* 1T 1 (M)

for i =F I, in 1T 1 (N) [96].

construction

,g-""N,h

range? Iblished for

Po

a harmonic

on K by a

K (e.g., that

(6.3) Problem. If M, g and N. h are compact and real ana(ytic, is the space H(M, N)

triangulable?

The answer is yes if Riemh ,.;;;; 0, as in (6.2). Also, the space of holomorphic maps be­

tween compact complex manifolds is an analytic space [87], and hence triangulable.

(6.4) Problem. Under what conditions can we deform the components of C(M, N) into H(M, N)? (6.5) It is well known that every compact Riemannian manifold N, h has a nontrivial closed geodesic.

(6.6) Problem. Characterize those manifolds N, h for which there exists a nontrivial harmonic map ¢: S2 -+ IV. As we saw in (9.12), such a map is a minimal branched immersion. When the univer­ sal cover Fl of N is noncontractible, such a map exists [53]. On the other hand, if RiemJli ,.;;;; 0, then every harmonic map S2 -+ N is constant [20]. Another question analogous to one for closed geodesics:

harmonic )nic map

'1

(6.7) Problem. For a given metric h on S3, are there at least 4 geometrically distinct

harmonic maps? Are there 4 such maps which are embeddings? ADDED IN PROOF. For the existence of one embedding, see: F. R. Smith, On the

finite energy,

existence of embedded minimal 2-spheres in the 3-sphere, endowed with an arbitrary metric,

anabgous as­

Ph. D. Thesis, Melbourne.

See [18; 10.15,

(6.8) Problem. Find topological restrictions on a manifold N. h to insure the exis­ tence of infinitely many harmonic maps S2 -+ N with distinct images? (6.9) As we have seen in (5.11) of Part I, if M is a product of spheres of dimensions ~ 3, then there are maps ¢: M -+ M homotopic to the identity with arbitrarily small ener­

ff measure of K. . 1. See also

gies. N. Koiso posed the following

(6.10) Problem [105]. Characten'ze those compact mamfolds M with that property; also, characten'ze their homotopy types. ; zero. For

.(jj, and by

ADDED IN PROOF. Min-Oo has shown that this property is satisfied by all simply con­ nected compact Lie groups, a key ingredient in the proof being that the cut-locus of a point is of codirnension at least three (Maps of minimum energy from compact simply connected

Lie groups). locally com-

(6.11) Results of [21] show that a closed orientable surface M of odd genus p ;;;;;. 3 carries two metrics go and gl relative to which there is a harmonic map M, go ~ S2 of de­ gree p, but no such map M, gl -> S2. Similarly:

76

JAMES EELLS AND LUC LEivlAIRE

(6.l2) Problem. Give an example of a harmonic map rp: il{ g --- N, h between com­ pact manifolds such that the metric h can be deformed into a metric hI such that rp is not homotopic to a harmonic map M, g --- N, hI' (6.13) Problem. Is there a generic set of mettlcs II on 11i for each of which there exist infinite{v many geometrical{v distinct harmonic maps 52 --- N, h?

fok!

is ar

In the case of closed geodesics, see [76].

(6.l4) Calabi [9,83] has completely classified the harmonic maps 52 --- Rpn, real

projective n-space with its standard metric h of constant curvature. (6.15) Problem. Is that classification reflected when the metric h is perturbed? 7. Noncompact domains. (7.1) Of course, general existence questions arise when the domain M is noncompact­ a harmonic map being characterized as an extremal of £ for all compactly supported varia­ tions. See [56,100].

(7.2) Problem. Classify the harmonic maps ¢: R m --- Rn which have maximal rank almost everywhere. If m = 1, those maps are affine. If m == 2 nand ¢ is injective, then again ¢ is af­ fine. If m = 2 and n = 3, the only known (nonplanar) injections are the catenoid and the helicoid. There are many immersions. Calabi has asked (7.3) Problem. Does there exist a nonplanar harmonically embedded (or properly immersed) surface in R3 with no tangent plane passing through the origin? (7.4) Problem. Is there a harmonic map from R2 to the hyperbolic plane H2 of rank 2 almost everywhere?

dim tion

ham diffc beh,a (pan

shou

Certainly such a map ¢ must have £(¢) == 00. Furthermore, ¢ cannot have bounded dilatation. (7.5) Problem. R. Osserman [103] has asked for a classification of the injective

If k Llonj

{z E C: PI < /z/ < P2} with PI > 0 and E == harmonic maps : A --- E, where A {z E C: 0 < Iz/ < I}. Nitsche has exhibited an interesting example. (7.6) Problem. Let M, g be a complete noncompact manifold with dim M;;;' 3 and Riemg ;;;. O. Under what conditions on N, h can we conclude that a harmonic map

mani

¢: M, g --- N, h with maximal rank;;;' 3 must have £(.d¢ = "d¢ for" E R, from the viewpoint of geometrical interpretations. It is important to keep in mind that the operator I::>. itself depends on ¢, so that we should not think of this problem as looking for a spectral decomposition of a fixed operator. (8.7) A polyharrnonic map of order k is an extremal of

: bounded

njecr've dB

If k > m12, then Fk satisfies Condition (C) of Palais-Smale; therefore, there is a polyhar­ monic map of order k in every homotopy class. (8.8) Problem. Study the existence of polyharmonic maps in the critical dimension

m = 2k. More precisely, what are the existence and nonexistence results analogous to those

'lie map

for harmonic maps in dimension 2? (8.9) The Plateau problem requires an extremal coboundary for a given closed sub­ manifold, without specifying its topological type. That suggests the following:

;6] .

Problem [90]. Given a compact orientable Riemannian manifold N and an integer m, consider pairs (¢, M), where M is a compact oriented Riemannian m-manifold with volume I

larmonic map

finite energy.

and ¢: M - N a smooth map. Say that two such pairs (¢o. Mo) and (¢)' M) bordant if there is

:ensions of

nons and

(1) a compact oriented Riemannian (m + i)-manifold W whose oriented boundary aW = M) - Mo has Riemannian structure that of Mo' M) ; (2) a smooth map ¢: W - N such that

.w;;:;:: 3 and

¢I A1k

a harmonic

= ¢k

for k

= 0,

are co­

1.

1t certain

That cobordism is an equivalence relation on those pairs; and the energy functional is de­

18, § 11].

fined on pairs.

78

JAMES EELLS AND LUC LEMAIRE

(8.l0) Problem. Under what conditions is there an extremal pair in a given cobordism class? (8.11) Dropping all orientability assumptions in (8.9), we can formulate an analogous problem in homology with Z2 -coefficients:

A class J.1. E Hm(N; Z2) is realized by a smooth map rf;: M -+ N of a compact m­ manifold M if the induced homomorphism 9*: Hmf.M; Z2) -+ Hm(N, Z2) carries the funda­ mental class of M onto J.1.. Thorn [114, Chapter III, 3] has shown that every class J.1. is so realized.

(8.12) Problem. When is a class J.1. E Hm(N; Z2) realized by a harmonic map 9: M, g -+ N, h of some M, g? (8.13) Problem. Develop a decent theory of harmonic maps between Riemannian piecewise linear mamfolds. Such spaces have canonically defined Lipschitz structures, and piecewise linear maps between them are Lipschitz. Stochastic Riemannian geometry might provide an interesting approach; see Part I (2.34).

fel

CO 1;:

21 tur

Ap SOl

din: Hol

tial! vo1. sc.'~

Nuc

419

r

ven cobordism ;;':1

analogous

npact m­ nes the funda­ ;lass 11 is so Bibliography for Part I ~onic

map

:iemannian linear maps :m interesting

1. N. Aronszajn, A unique continuation theorem for solutions of elliptic partial dif­ ferential equations or inequalities of second order, 1. Math. Pures Appl. 36 (1957), 235-249. 2. N. Aronszajn, A. Krzywicki and J. SZarski, A unique continuation theorem for ex­ terior differential forms on Riemannian manifolds, Ark. Mat. 4 (l962), 417-453.

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